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authorPrefetch2023-05-13 15:42:47 +0200
committerPrefetch2023-05-13 15:42:47 +0200
commit7c412050570ef229dd78cbcffbf80f23728a630d (patch)
treebbb70d82673ddaddb1b259d8b7a4991993f61bc5 /source/know/concept/ficks-laws
parent9d9693af6fb94ef4404a3c2399cb38842e5ca822 (diff)
Improve knowledge base
Diffstat (limited to 'source/know/concept/ficks-laws')
-rw-r--r--source/know/concept/ficks-laws/index.md21
1 files changed, 13 insertions, 8 deletions
diff --git a/source/know/concept/ficks-laws/index.md b/source/know/concept/ficks-laws/index.md
index b205af9..8d5da7d 100644
--- a/source/know/concept/ficks-laws/index.md
+++ b/source/know/concept/ficks-laws/index.md
@@ -14,6 +14,7 @@ A diffusion process that obeys Fick's laws is called **Fickian**,
as opposed to **non-Fickian** or **anomalous diffusion**.
+
## Fick's first law
**Fick's first law** states that diffusing matter
@@ -21,13 +22,14 @@ moves from regions of high concentration to regions of lower concentration,
at a rate proportional to the difference in concentration.
Let $$\vec{J}$$ be the **diffusion flux** (with unit $$\mathrm{m}^{-2} \mathrm{s}^{-1}$$),
-whose magnitude and direction describe the "flow" of diffusing matter.
+whose magnitude and direction describes the "flow" of diffusing matter.
Formally, Fick's first law predicts that the flux
is proportional to the gradient of the concentration $$C$$ (with unit $$\mathrm{m}^{-3}$$):
$$\begin{aligned}
\boxed{
- \vec{J} = - D \: \nabla C
+ \vec{J}
+ = - D \: \nabla C
}
\end{aligned}$$
@@ -43,6 +45,7 @@ but they say more about those systems
than about diffusion in general.
+
## Fick's second law
To derive **Fick's second law**, we demand that matter is conserved,
@@ -65,8 +68,7 @@ $$\begin{aligned}
= - \int_V \nabla \cdot \vec{J} \dd{V}
\end{aligned}$$
-For comparison, we differentiate the definition of $$M$$,
-and exploit that the integral ignores $$t$$:
+For comparison, we can also just differentiate the definition of $$M$$ directly:
$$\begin{aligned}
\dv{M}{t}
@@ -74,7 +76,8 @@ $$\begin{aligned}
= \int_V \pdv{C}{t} \dd{V}
\end{aligned}$$
-Both $$\idv{M}{t}$$ are equal, so stripping the integrals leads to this **continuity equation**:
+Above, we have two valid expressions for $$\idv{M}{t}$$,
+which must be equal, so stripping the integrals leads to this **continuity equation**:
$$\begin{aligned}
\pdv{C}{t}
@@ -101,6 +104,7 @@ $$\begin{aligned}
\end{aligned}$$
+
## Fundamental solution
Fick's second law has exact solutions for many situations,
@@ -110,11 +114,12 @@ where the initial concentration $$C(x, 0)$$ is
a [Dirac delta function](/know/concept/dirac-delta-function/):
$$\begin{aligned}
- C(x, 0) = \delta(x - x_0)
+ C(x, 0)
+ = \delta(x - x_0)
\end{aligned}$$
-According to Fick's second law,
-the concentration's time evolution of $$C$$ turns out to be:
+By solving Fick's second law with this initial condition,
+$$C$$'s time evolution turns out to be:
$$\begin{aligned}
H(x - x_0, t)